MOTIVATION FOR EXTENDING THE CONCEPT OF A VECTOR TO INCLUDE OBJECTS OF AN ARBITRARY NATURE. GENERALIZATION OF THE CONCEPT OF A VECTOR SPACE 

If we consider the vectors of regular two and three dimensional space we observe that under the operations of addition and multiplication by numbers they form closed systems -- closed systems that we call groups and subgroups, spaces and subspaces. Each space and subspace has a dimension given by the number of linearly independent vectors required to span it. Not only do linear combinations of vectors of two and three dimensional space separate out into spaces and subspaces in this way but so, too, do vectors of Euclidean n-dimensional space. In Matrix Theory we study vectors of Euclidean n-dimensional space. However, in other areas of mathematical study we also encounter this same phenomenon where “things” of various kinds, under addition and multiplication by numbers, form closed systems, groups and subgroups, spaces and subspaces in the same way as do the vectors of two and three dimensional space. In the study of integral equations, differential equations, polynomials, infinite series and infinite sequences, etc. we encounter the same thing. Because of this it has been found useful to generalize the concept of a vector to include “objects” of an arbitrary nature (for which addition and multiplication by a number make sense) and to generalize the entire concept of a vector space, putting it on an axiomatic foundation, separating it from its tie to Euclidean n-dimensional space.